psychologywikiaorg-20200213-history
Schr
In physics, especially quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics. In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only atomic and subatomic systems, atoms and electrons, but also macroscopic systems, possibly even the whole universe. The equation is named after Erwin Schrödinger, who constructed it in 1926. Schrödinger's equation can be mathematically transformed into Heisenberg's matrix mechanics, and into Feynman's path integral formulation. The Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem which is not as severe in Heisenberg's formulation and completely absent in the path integral. The Schrödinger equation The Schrödinger equation takes several different forms, depending on the physical situation. This section presents the equation for the general case and for the simple case encountered in many textbooks. http:// For a general quantum system: : :: where : :* is the imaginary unit :* is the wave function, which is the probability amplitude for different configurations of the system. :* is the reduced Planck's constant (often normalized to 1 in natural units). :* is the Hamiltonian operator. http:// For a single particle in three dimensions: : :: where : :* is the particle's position in three-dimensional space, :* is the wavefunction, which is the amplitude for the particle to have a given position r''' at any given time t. :* is the mass of the particle. :* is the time independent external with respect to the particle potential energy of the particle at each position '''r (see Self-action in a system of elementary particles). :* is the Laplace operator. Historical background and development Einstein interpreted Planck's quanta as photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, a mysterious wave-particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in relativity, it followed that the momentum of a photon is proportional to its wavenumber. DeBroglie hypothesized that this is true for all particles, for electrons as well as photons, that the energy and momentum of an electron are the frequency and wavenumber of a wave. Assuming that the waves travel roughly along classical paths, he showed that they form standing waves only for certain discrete frequencies, discrete energy levels which reproduced the old quantum condition. Following up on these ideas, Schrödinger decided to find a proper wave equation for the electron. He was guided by Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system--- the trajectories of light rays become sharp tracks which obey an analog of the principle of least action. Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves. This is what Schrödinger did, and a modern version of his reasoning is reproduced in the next section. The equation he found is (in natural units): : :: Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , moving in a potential well, V', created by the positively charged proton. This computation reproduced the energy levels of the Bohr model. But this was not enough, since Sommerfeld had already seemingly correctly reproduced relativistic corrections. Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential: : :: He found the standing-waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin with a lover. While there, Schrödinger decided that the earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. He put together his wave equation and the spectral analysis of hydrogen in a paper in 1926. The paper was enthusiastically endorsed by Einstein, who saw the matter-waves as the visualizable antidote to what he considered to be the overly formal matrix mechanics. The Schrödinger equation tells you the behaviour of , but does not say what ''is. Schrödinger tried unsuccessfully, in his fourth paper, to interpret it as a charge density. In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted as a probability amplitude. Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities; like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory, Schrödinger was never reconciled to the Copenhagen interpretation. Derivation http:// http:// :(1) The total energy E'' of a particle is : :: :::This is the classical expression for a particle with mass ''m where the total energy E'' is the sum of the kinetic energy, , and the potential energy ''V. The momentum of the particle is '''p, or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well. :Note that the energy E'' and momentum '''p' appear in the following two relations: :(2) Einstein's light quanta hypothesis of 1905, which asserts that the energy E'' of a photon is proportional to the frequency ''f of the corresponding electromagnetic wave: : :: :::where the frequency f'' of the quanta of radiation (photons) are related by Planck's constant ''h,:::and is the angular frequency of the wave. :(3) The de Broglie hypothesis of 1924, which states that any particle can be associated with a wave, represented mathematically by a wavefunction Ψ, and that the momentum p''' of the particle is related to the wavelength λ of the associated wave by: : :: :::where is the wavelength and is the wavenumber of the wave. :Expressing '''p and k''' as vectors, we have :: http:// Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor: : :: and to realize that since : :: then : :: and similarly since : :: and : :: we find: : :: so that, again for a plane wave, he obtained: : :: And by inserting these expressions for the energy and momentum into the classical formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential '''V: : :: http:// The particle is described by a wave; the frequency is the energy E of the particle, while the momentum p is the wavenumber k. Because of special relativity, these are not two separate assumptions. : :: The total energy is the same function of momentum and position as in classical mechanics: : :: ::: where the first term T''(''p) is the kinetic energy and the second term V''(''x) is the potential energy. Schrödinger required that a Wave packet at position x with wavenumber k will move along the trajectory determined by Newton's laws in the limit that the wavelength is small. Consider first the case without a potential, V=0. : :::: So that a plane wave with the right energy/frequency relationship obeys the free Schrödinger equation: : :: and by adding together plane waves, you can make an arbitrary wave. When there is no potential, a wavepacket should travel in a straight line at the classical velocity. The velocity v of a wavepacket is: : :: which is the momentum over the mass as it should be. This is one of Hamilton's equations from mechanics: : :: after identifying the energy and momentum of a wavepacket as the frequency and wavenumber. To include a potential energy, consider that as a particle moves the energy is conserved, so that for a wavepacket with approximate wavenumber k at approximate position x the quantity : :: must be constant. The frequency doesn't change as a wave moves, but the wavenumber does. So where there is a potential energy, it must add in the same way: : :: This is the time dependent Schrödinger equation. It is the equation for the energy in classical mechanics, turned into a differential equation by substituting: : :: Schrödinger studied the standing wave solutions, since these were the energy levels. Standing waves have a complicated dependence on space, but vary in time in a simple way: : :: substituting, the time-dependent equation becomes the standing wave equation: : :: Which is the original time-independent Schrödinger equation. In a potential gradient, the k-vector of a short-wavelength wave must vary from point to point, to keep the total energy constant. Sheets perpendicular to the k-vector are the wavefronts, and they gradually change direction, because the wavelength is not everywhere the same. A wavepacket follows the shifting wavefronts with the classical velocity, with the acceleration equal to the force divided by the mass. An easy modern way to verify that Newton's second law holds for wavepackets is to take the Fourier transform of the time dependent Schrödinger equation. For an arbitrary polynomial potential this is called the Schrödinger equation in the momentum representation: : :: ::: The group-velocity relation for the fourier transformed wave-packet gives the second of Hamilton's equations. : :: ::: Versions There are several equations which go by Schrödinger's name: http:// This is the equation of motion for the quantum state. In the most general form, it is written: : :: Where is a linear operator acting on the wavefunction . takes as input one and produces another in a linear way, a function-space version of a matrix multiplying a vector. For the specific case of a single particle in one dimension moving under the influence of a potential V. : :: and the operator H can be read off: : :: it is a combination of the operator which takes the second derivative, and the operator which pointwise multiplies by V(x). When acting on it reproduces the right hand side. For a particle in three dimensions, the only difference is more derivatives: : :: and for N particles, the difference is that the wavefunction is in 3N-dimensional configuration space, the space of all possible particle positions. :This last equation is in a very high dimension, so that the solutions are not easy to visualize. http:// This is the equation for the standing waves, the eigenvalue equation for H. In abstract form, for a general quantum system, it is written: : :: For a particle in one dimension, : :: But there is a further restriction--- the solution must not grow at infinity, so that it has a finite L^2-norm: : :: For example, when there is no potential, the equation reads: : :: which has oscillatory solutions for E>0 (the C's are arbitrary constants): : :: and exponential solutions for E<0 : :: The exponentially growing solutions have an infinite norm, and are not physical. They are not allowed in a finite volume with periodic or fixed boundary conditions. For a constant potential V the solution is oscillatory for E>V and exponential for E